Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 48 (2007), No. 2, 443-467.
Virtual Points and Separating Sets
in Spherical Circle Planes
Burkard Polster
Günter F. Steinke
School of Mathematical Sciences, Monash University
Vic 3800, Australia
e-mail: burkard.polster@sci.monash.edu.au
Department of Mathematics, University of Canterbury
Christchurch, New Zealand
e-mail: G.Steinke@math.canterbury.ac.nz
Abstract. In this paper we introduce and investigate virtual points of
spherical circle planes and use them to construct several new types of
separating sets of circles in such geometries. Two spherical circle planes
combine very naturally into another spherical circle plane if they share
such a separating set of circles.
MSC2000: 51B10, 51H15
Keywords: spherical circle planes, flat Möbius planes, separating sets,
virtual points
1. Introduction
Associated with a strictly convex topological sphere in real projective three-space
is the circle geometry of nontrivial plane sections of the sphere. For example, the
classical spherical circle plane is the geometry of nontrivial plane sections of the
unit sphere and, in this special case, the nontrivial plane sections are just the
Euclidean circles contained in the unit sphere. The set of circles corresponding to
the planes through a point in space gives rise to an interesting subgeometry of the
circle plane. This subgeometry is a flat affine plane if the point is on the sphere, a
double cover of a flat projective plane if the point is an inner point of the sphere
and a double cover of an R2 -plane if the point is an outer point of the sphere.
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General spherical circle planes usually do not come with a surrounding projective space. To generalize the subgeometries that correspond to points off the
sphere, we introduce virtual inner and outer points of spherical circle planes that
can play the roles of the respective inner and outer points of a strictly convex
topological sphere in real projective three-space. Here a virtual inner or outer
point of a spherical circle plane is a special involutory homeomorphism of the
point space of the circle plane to itself. Its associated geometry of fixed circles is
of the same type as the geometry of fixed (Euclidean) circles of the bundle involution associated with an inner or outer point of the unit sphere. The connection
with what we said before is established by the fact that the geometry of fixed
circles of a bundle involution associated with a point coincides with the nontrivial
plane sections of the sphere with the planes containing this point.
Two points in real projective three-space separate their connecting line into
two connected components. Every plane in this space either contains the whole
line or intersects it in exactly one point. This implies that the set of planes
which contain one or both of the distinguished points also separates the set of
planes of real projective three-space into two connected components. In turn this
implies that the set of planes which contain one or both of the distinguished points
corresponds to a set of circles of the circle plane that separates the circle set of
this geometry into two connected components. Of course, this set of circles is
just the union of the circle sets of the two subgeometries associated with the two
points.
The main focus of this paper is to investigate similar separating sets of circles
in general spherical circle planes constructed from pairs of actual and virtual
points of such planes. These special sets of circles do not only separate the circle
sets of the spherical circle planes that they are contained in topologically, but also
geometrically: given two spherical circle planes that share such a separating set,
it is possible to recombine the separating set and one associated component of
circles each taken from the two spherical circle planes, into the circle set of a new
spherical circle plane.
This paper is organized as follows. In Section 2 we give a brief introduction
to spherical circle planes and separating sets, introduce virtual inner and outer
points of general spherical circle planes and derive some basic properties of these
virtual points. In Section 3 we introduce seven new types of separating sets in
spherical circle planes.
We only note that separating sets of circles similar to the ones investigated in
this paper have been shown to exist in other types of geometries on surfaces, such
as flat projective planes, cylindrical circle planes and toroidal circle planes; see [1],
[2], [3] and, in particular, [4, Chapters 2.7.10, 3.3.4, 4.3.1, 4.3.2, 5.3.3 and 7.3.2]
for more detailed information. Furthermore, see [5, Example 31.25(b)] and [6] for
information about the closely related Moulton constructions which are based on
separating sets contained in the point sets of flat linear spaces.
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2. Spherical circle planes and separating sets
In this section we collect some basic information about spherical circle planes, and
give definitions of virtual points and separating sets.
Axioms for spherical circle planes
A spherical circle plane is a point-circle geometry whose point set is homeomorphic
to the unit sphere S2 and whose circles are simply closed curves embedded in the
point set. Furthermore, every spherical circle plane satisfies the following Axiom
of Joining:
• Three points are contained in exactly one circle.
A spherical circle plane is nested if it satisfies the following additional Axiom of
Touching:
• Given two points and a circle through at least one of the points, there is
exactly one circle that contains both points and touches the given circle,
that is, coincides with this circle or intersects it in only one point.
Nested spherical circle planes are also known as flat Möbius planes and are Möbius
planes in the usual incidence-geometric sense.
Ovoidal planes
Consider the geometry of nontrivial plane sections of a strictly convex topological
sphere in real projective three-space. This geometry is an ovoidal spherical circle
plane. It is nested if and only if the surface is differentiable, that is, if the surface
has a unique tangent plane at every single one of its points. The spherical circle
plane associated with the sphere S2 is the classical spherical circle plane. Its circles
are just the Euclidean circles on S2 . Many non-ovoidal spherical circle planes are
known.
Topological geometries
Spherical circle planes are automatically topological geometries in the sense that
the connecting circle of three points depends continuously on the three points, and,
in the case of a nested plane, the touching circle depends continuously on the given
two points and circle. More precisely, with respect to the natural topology on the
point set and the topology induced by the Hausdorff metric on the circle set,
the geometric operations are continuous operations on their respective domains
of definition. By the circle space of a spherical circle plane we mean the circle
set provided with the topology induced by the Hausdorff metric. The circle space
is a connected, three-dimensional locally Euclidean space. In fact, it is known
that the circle space of a flat Möbius plane is homeomorphic to a real projective
3-space from which one point has been removed.
Here is one more important fact which establishes spherical circle planes as
very well-behaved topological geometries. Given a spherical circle plane with
point set P , any two of its circles c and d having two points in common intersect
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transversally in these two points. This means that parts of c are contained in
both connected components of P \ d. On the other hand, any two simply closed
curves c and d on the sphere intersecting in exactly one point clearly touch in this
point. This means that c minus this point is contained in one of the connected
components of P \ d.
See [4, Chapter 3] for a comprehensive summary of results about spherical
circle planes.
Separating sets
From now on we will assume, without loss of generality, that all spherical circle
planes have the unit sphere S2 as point set. We will describe various types of
separating sets of spherical circle planes. Here a separating set S is a subset of a
circle space C satisfying the following axioms:
(S1) The complement C \ S has two path-connected components C + and C − .
The circles in C + and C − can be characterized in terms of S.
(S2) Given three points r, s, and t on the sphere, it suffices to look at the position
of these three points with respect to the separating set to decide whether
the connecting circle is contained in S, C + , or C − .
(S3) If we are dealing with a flat Möbius plane, it suffices to look at the position
of two points r and s, and a circle c that contains r, to decide whether the
circle d through s that touches c in r is contained in S, C + , or C − .
Now assume that a certain set of circles S is a separating set in two different
spherical circle planes. Then it follows from axiom S2 that S together with C +
taken from the first circle plane and C − taken from the second circle plane forms
the circle set of a new spherical circle plane. Furthermore, if the two planes we
started with are flat Möbius planes, then, as a consequence of axiom S3, the new
plane will be a flat Möbius plane as well.
Let us switch back to the spatial model. Fix a line l and two points p and q
on this line. Clearly, the set l \ {p, q} has two connected components, and every
plane in the space either contains the line or intersects the line in exactly one
point. This means that the set of planes containing p or q also gives rise to a
set of circles that separates the circle space of the classical spherical circle plane
into two connected components. Depending on the position of the two points and
their connecting line with respect to the sphere, we distinguish eleven cases, as
indicated in Figure 1.
Note that in the diagrams a solid point, an open inner point of the circle,
and an open outer point of the circle represents a point on, an inner point, and
an outer point of the sphere, respectively. The tangents to the sphere through
an outer point touch the sphere in points that form a circle which we will call
the horizon of the outer point. In the following, these horizons will play a special
role and, in the diagrams, these special circles are represented by dashed lines. In
diagram 5a the horizon of the outer point is also a circle associated with the inner
point. In diagram 6d the horizon associated with one of the two outer points is
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1
5a
2
5b
3
6a
4b
4a
6b
6c
6d
Figure 1. Eleven different positions of a pair of points with respect to a sphere
also one of the circles associated with the respective other outer point. Note that
the cases captured by the diagrams present the complete picture in the classical
case and that other conceivable scenarios such as a ‘mixed’ type in between types
6c and 6d do not occur in this case.
To be able to define virtual inner and outer points that can play the role
of inner and outer points in general spherical circle planes, consider the bundle
involution i : S2 → S2 associated with a point r of the space not contained in S2 ;
see Figure 2.
Figure 2. Bundle involutions associated with inner and outer points of S2
Let s be a point of S2 . Then i fixes s if the line connecting r and s is a tangent of the
sphere. Otherwise, it interchanges s with the unique second point of intersection
of the connecting line with the sphere. The bundle involution associated with r
is a homeomorphism. If r is an inner point of the sphere, it is
(I) fixed-point-free and orientation-reversing.
If r is an outer point of the sphere, it is
(O) orientation-reversing and the set of fixed points fix(i) of the involution is a
circle of the circle plane.
Furthermore, if FIX(i) denotes the set of circles that are globally but not pointwise
fixed by i, then in both cases
(B) two points of the spherical circle plane that are not interchanged by the
involution are contained in exactly one circle in FIX(i).
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Note that the circles that are globally fixed but not pointwise fixed by a bundle
involution associated with a point r are just the circles that correspond to the
planes through r. The set of fixed points of an outer involution forms the horizon
that we mentioned earlier.
Now, consider an arbitrary spherical circle plane with point set S2 . Let i :
2
S → S2 be an involutory homeomorphism. If i satisfies conditions (I) and (B)
above, we call it an inner involution of the circle plane. If i satisfies conditions
(O) and (B) above, we call it an outer involution. Note that, unlike in the classical
case, an inner or outer involution need not be an automorphism of the spherical
circle plane.
In the next section we will describe seven different types of separating sets in
general spherical circle planes that correspond to all those cases shown in Figure 1
in which the connecting line of the two points intersects the sphere in two points.
These are cases 1, 2, 3, 4b, 5a, 5b, 6a. In the definitions of these separating sets
inner and outer involutions play the role of inner and outer points. Therefore, we
will also refer to inner and outer involutions as virtual inner and outer points.
Examples. Bundle involutions can also be defined for ovoidal spherical circle
planes associated with convex topological spheres that are not Euclidean spheres.
Those bundle involutions associated with inner points always give examples of
inner involutions. A bundle involution associated with an outer point corresponds
to an outer involution only if its set of fixed points is a circle of the circle plane (in
general this is not the case). Usually bundle involutions of ovoidal circle planes are
not automorphisms of these planes. The eleven cases shown in Figure 1 only give
a complete picture in the classical case. In non-classical ovoidal circle planes a
number of additional configurations can occur in the case of two outer points. For
example, it is possible that two different outer involutions of an ovoidal spherical
circle plane have the same set of fixed points. Here is how you construct such
a spherical circle plane: Join two copies of a spherical cap along their circular
boundaries to create a convex topological sphere. If the cap you started with is
not a hemisphere, then the resulting topological sphere has only one symmetry
axis, and the bundle involutions associated with infinitely many points on this
symmetry axis have the circle at which the two caps are joined together as fixed
circle.
It is also easy to construct inner and outer involutions that are inner and outer
involutions of two different ovoidal spherical circle planes. Consider, for example,
two different strictly convex topological spheres O1 and O2 , one nested inside the
other, and the associated ovoidal spherical circle planes S1 and S2 . Furthermore,
let p be a point in the interior of both. Projecting the circles of S2 through p onto
O1 yields a spherical circle plane S20 with point set O1 isomorphic to S2 . Clearly,
the bundle involution with respect to p and O1 is an inner involution of both S1
and S20 . We leave constructing two spherical circle planes that share an outer
involution as an exercise.
Examples of outer involutions in nonovoidal spherical circle planes include
those involutions of the so-called semi-classical Möbius planes Mid,h (see [4, Chap-
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449
ter 3.3.3]) that correspond to the map (x, y) 7→ (−x, y). Furthermore, if h 6= id,
then these outer involutions are not automorphisms of Mid,h . These planes also
admit each reflection (x, y) 7→ (x, 2t − y) at the horizontal line y = t as an outer
involution. Another class of nonovoidal spherical circle planes that admit outer
involutions are certain Ewald planes (see [4, Chapter 3.3.2]). Starting from an
oval O in the Euclidean plane that has a line l as a symmetry axis one first dilates O from one point with all possible positive factors and then translates the
resulting ovals in all possible directions. These spherical circle planes then admit
reflections about each line parallel to l as outer involutions.
Basic properties of inner and outer involutions
Lemma 2.1. (Inner involutions) Let i be an inner involution of a spherical circle
plane with circle set C. Then the following hold:
1. The circles through a point p that are contained in FIX(i) are exactly the
circles through p and i(p).
2. Two distinct circles in FIX(i) have exactly two points in common. These
two points get interchanged by i.
3. Let c ∈ C \ FIX(i). Then c is disjoint from its image i(c) under the involution. (Note that i(c) is not necessarily a circle in C.)
4. Let r, s, i(r), and i(s) be four distinct points on a circle c in FIX(i). Then
r and i(r) are contained in different connected components of c \ {s, i(s)}.
5. It is possible to reconstruct i from FIX(i).
Proof. Remember that i is fixed-point-free and orientation-reversing. This means
that, restricted to one of its fix-circles, it is an orientation-preserving involutory
homeomorphism.
(1) By definition, a circle that contains p and is contained in FIX(i) also contains
i(p). On the other hand, let c be a circle through p and i(p), and let q be a third
point on this circle. Since i satisfies condition (B), the set FIX(i) contains a circle
d connecting p and q. This circle also contains i(p). Because there is exactly one
circle in C containing p, i(p), and q, the circles c and d coincide. Hence every
circle through p and i(p) is contained in FIX(i).
(2) As a consequence of the axiom of joining, two circles in C intersect in 0, 1,
or 2 points. Let c and d be two distinct circles in FIX(i). If the two circles were
to intersect in just one point, this point would be a fixed point of i. Since i is
fixed-point-free this is not possible. Assume that c and d do not intersect. Then,
since the connected components of S2 \ c are interchanged, d and i(d) are distinct,
which is not the case. We conclude that c and d intersect in exactly two points.
Also, since i does not have any fixed points, we conclude that these two points
are interchanged by i.
(3) Let c ∈ C \ FIX(i). Assume that there is a point p that is contained in both
c and i(c). Then the same is true for i(p), and by (1), the circle c is contained in
FIX(i). This is a contradiction.
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(4) follows immediately from the fact that i restricted to c is a fixed-point-free
involution.
(5) Given a point p, consider any two circles in FIX(i) through p. Then, by
Lemma 2.1.2., the point i(p) is necessarily the second point of intersection of the
two circles. This completely determines i.
Lemma 2.2. (Outer involutions) Let i be an outer involution of a spherical circle
plane with circle set C. Then the following hold:
1. Let p be a point that is not fixed by i. Then the circles through p that are
contained in FIX(i) are exactly the circles in the pencil of circles through p
and i(p).
2. Let c ∈ C \ FIX(i). Then c ∩ i(c) = c ∩ fix(i).
3. Let r, s, i(r), and i(s) be four distinct points on a circle c in FIX(i). Then
r and i(r) are contained in the same connected component of c \ {s, i(s)}.
4. The circles in FIX(i) that contain a point of fix(i) all touch at this point. If
we are dealing with a flat Möbius plane, then these circles form a touching
pencil of circles.
5. It is possible to reconstruct i from FIX(i).
We skip the proof of this result since it is very similar to the proof of Lemma 2.1.
The topology of the fixgeometries of inner and outer involutions
To figure out what FIX(i) looks like topologically, we consider the fixgeometries
associated with our special involutions. Depending on whether i is inner or outer,
the point set of this fixgeometry is S2 or S2 \ fix(i) with pairs of points that are
interchanged by the involution identified. Its line set consists of the elements of
FIX(i) which have been identified accordingly.
A flat projective plane is a projective plane whose point set is homeomorphic
to the real projective plane and whose lines are subsets of this point set homeomorphic to the circle. The classical example of a flat projective plane is the
projective plane over the real numbers. An R2 -plane is a linear space whose
point set is homeomorphic to R2 and whose lines are subsets of this point set
homeomorphic to R which separate the point set into two open components. The
classical example of an R2 -plane is the Euclidean plane. Now the following result
is an easy corollary of property (B) shared by both inner and outer involutions.
Proposition 2.3. (Types of fixgeometries) The fixgeometry of an inner involution of a spherical circle plane is a flat projective plane. The fixgeometry of an
outer involution is an R2 -plane.
For example, the fixgeometry associated with an inner or outer point of the classical spherical circle plane is isomorphic to the classical flat projective plane or
the restriction of this plane to the interior of the unit circle, respectively.
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Now it is clear that FIX(i) is homeomorphic to the line set of the fixgeometry
of i. Using well-known properties of flat projective planes and R2 -planes (see, e.g.,
[4, Theorem 2.3.6]), we arrive at the following result:
Corollary 2.4. (Topology) Let i be an inner or outer involution.
1. If i is an inner involution, then FIX(i) is homeomorphic to the real projective
plane.
2. If i is an outer involution, then FIX(i) is homeomorphic to a Möbius strip
(without boundary).
3. As long as the points r and s are not interchanged by i, the unique circle
in FIX(i) that contains both points depends continuously on the position of
these points.
Combining two involutions or combining an actual point and an involution
Consider a point p on the unit sphere and a point q off the sphere in real projective
three-space; see again Figure 1. The connecting line of p and q touches the sphere
in exactly one point if the line touches the sphere at p. In this case q is necessarily
an outer point of the sphere. Otherwise, the line intersects the sphere in exactly
two points. Lemmas 2.1.1, 2.2.1 and 2.2.4 show how these facts generalize if we
replace q by an inner or outer involution of a spherical circle plane. We summarize
these results and supplement them by some topological interpretations that follow
immediately from Corollary 2.4.
Lemma 2.5. (One actual and one virtual point) Let p be a point on the sphere
and let q be an inner or outer involution of a spherical circle plane.
1. If p is not fixed by q, then the circles through p that are fixed by q are all
the circles through p and q(p). Topologically this set of circles is a nonseparating topological circle inside the surface formed by FIX(q). It is also a
non-separating circle inside the surface formed by all the circles containing
the point p (this surface is homeomorphic to a Möbius strip).
2. If p is fixed by q, then the circles through p that are fixed by q all touch at the
point p. Topologically this set of circles is homeomorphic to a non-separating
open interval inside the surface formed by FIX(q).
Consider two points p and q off the unit sphere in real projective three-space;
see again Figure 1. If their connecting line intersects the sphere in two points p0
and q 0 , then p0 , q 0 is the only pair of points on the sphere that gets interchanged
by both the bundle involutions associated with p and q. This fact generalizes as
follows:
Lemma 2.6. (Two virtual points) Let p be an inner involution and let q be either
an inner involution different from p or an outer involution. Or let p and q be outer
involutions such that fix(p) ∩ fix(q) = ∅. Then the following hold:
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1. There is a unique pair of points p0 and q 0 on the sphere that are interchanged
by both p and q.
2. The pencil of circles through p0 and q 0 is equal to FIX(p) ∩ FIX(q).
3. It is possible to reconstruct p and q from FIX(p) ∪ FIX(q).
4. Topologically FIX(p)∩FIX(q) is a simply closed non-separating curve in both
the surfaces formed by FIX(p) and by FIX(q).
Proof. (1) Note that pq, as a product of two orientation-reversing homeomorphisms of the sphere to itself, is a continuous orientation-preserving homeomorphism. As such it has at least one fixed point p0 . Hence pq(p0 ) = p0 , and thus
q(p0 ) = p(p0 ). Since p and q do not have a common fixed point, the points p0
and q 0 = q(p0 ) = p(p0 ) are distinct. We conclude that p0 and q 0 are interchanged
by both involutions.
Assume that there is a second pair of points p00 , q 00 on the sphere that is
interchanged by both involutions. Also, let us first assume that p0 , q 0 , p00 , q 00 are
not all contained in a circle. Let u be any point different from p0 , p00 , q 0 , q 00 and
make sure that u is fixed by neither p nor q. Then, by Lemmas 2.1.1 and 2.2.1, the
circle containing p0 , q 0 , u and the circle p00 , q 00 , u are both fixed by p and q. The two
circles intersect in u. Since u is not a fixed point of either involution, the circles
have to intersect in a second point u0 , and, consequently, both p and q interchange
u and u0 . This implies that the two involutions are equal off fix(p) ∪ fix(q) and
therefore, by continuity everywhere, which is not the case. If p0 , q 0 , p00 , q 00 are all
contained in a circle, then we can conclude as just now that both involutions
operate in the same way off this circle. However, by continuity, it then follows
that both involutions are also operating in the same way on this circle and are
therefore identical. We conclude that p0 , q 0 is the only pair of points that gets
interchanged by both involutions.
(2) Again, as a consequence of Lemmas 2.1.1 and 2.2.1, the pencil of circles through
p0 and q 0 is contained in FIX(p)∩FIX(q). If FIX(p)∩FIX(q) included circles other
than those in the pencil, intersecting one of these other circles with a suitably
chosen one in the pencil would give two further points that are interchanged by
both involutions, which is impossible.
(3) It also follows that the only pencil of circles in FIX(p) ∪ FIX(q) that cuts
FIX(p) ∪ FIX(q) into two connected halves is FIX(p) ∩ FIX(q). Therefore, FIX(p)
and FIX(q) can be reconstructed from FIX(p) ∪ FIX(q), and, in turn, the involutions p and q can be reconstructed from FIX(p) and FIX(q), by Lemmas 2.1.5
and 2.2.5.
(4) This is a simple consequence of the definition of the fixgeometry of one of our
involutions.
Problems. Instead of intersecting the sphere in two points, the connecting line
of two outer points of the sphere may only touch the sphere in one point or miss
the sphere altogether. This is the case iff the corresponding horizons touch in
a point or intersect in two points. Also, as we have already pointed out in our
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453
discussion of inner and outer involutions of non-classical ovoidal circle planes, it is
possible that the horizons of two outer involutions coincide. Anyway, in the case
that the two pointwise fixed circles of two outer involutions intersect, it seems
difficult to determine how many pairs of points there are that are interchanged
by both involutions. Consequently, we can also not say much about the structure
of the set of circles that is fixed by both involutions.
3. Seven new separating sets
In this section we establish the existence of seven new types of separating sets
associated with two actual or virtual points p and q of a spherical circle plane.
These seven types correspond to all those cases shown in Figure 1 in which the
connecting line of the two points intersects the sphere in two points. These are
cases 1, 2, 3, 4b, 5a, 5b, and 6a. At the end of this section we will discuss the
main obstacles in the way of also generalizing those cases in which the connecting
line of the two points only touches or misses the sphere.
If C is the circle set of our spherical circle plane, let C(p) denote the set of
circles through p if p is a point of the sphere, or C(p) := FIX(p) if p is an inner or
outer involution. Furthermore C ∩ (p, q) := C(p) ∩ C(q), and the separating set is
C ∪ (p, q) := C(p) ∪ C(q). We will abbreviate C ∩ (p, q) and C ∪ (p, q) by C ∩ and C ∪ ,
whenever this will not lead to confusion.
In all cases that we will be considering in the following, the two path-connected
components of C \ C ∪ will be denoted by C + (p, q) and C − (p, q), or, for short, C +
and C − . We describe the components C + and C − in terms of C ∪ in the box at
the beginning of every subsection. Of course, it suffices to describe just one of
the components, for example C + , since C − = C \ (C + ∪ C ∪ ). Note that C(p) is
a surface which is homeomorphic to a Möbius strip if p is a point of the sphere
or an outer involution, or the real projective plane if p is an inner involution; see
Corollary 2.4. As we have shown in the previous section, in all of the cases under
consideration C ∩ (p, q) is a pencil of circles through two points. So, the picture
to keep in mind is that of C being a three-dimensional space that is cut by the
two-dimensional space C ∪ into the two three-dimensional chunks C + and C − .
Important additional requirement: For those separating sets C ∪ (p, q) that involve
outer involutions we will also always require that the fixed-point sets of all involved
outer involutions are circles of the circle planes that we are dealing with. To
understand why we have to require this extra alignment note the following open
problem:
Open Problem. Given two spherical circle planes C and D, and an outer involution i of C such that FIX(i) is contained in D, is i also an outer involution
of D, that is, is fix(i) also a circle of D?
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Procedure for establishing the existence of the new separating sets
In the following, we will work our way through the different types of separating
sets/candidates in the order set out in Figure 1. As we proceed, the proofs will get
more involved. However, since there are also lots of similarities in what exactly
has to be checked for each separating set, we will be able to skip more and more
details as we go along. Therefore, if you intend to understand the individual facts
that need to be verified for one of the types of separating sets under discussion,
it is important that you work your way through all of the discussion of the types
that precede the one that you are interested in.
In order to check that a set C ∪ is really a separating set, we have to show that
it satisfies axioms S1, S2, and S3. In every single case we verify this by proceeding
in the order S2, S3, S1.
Note that the descriptions of C + and C − and our proofs will often make use
of the involutions that entered into the original construction of C ∪ . This only
makes sense because, as we have shown in Lemmas 2.1.5, 2.2.5 and 2.6.4, these
involutions can be reconstructed from C ∪ .
Whenever we are dealing with an outer involution i, the two connected components of S2 \ fix(i) will be denoted by H + (i) and H − (i). We will specify in every
single case which half carries which name.
The proofs will be accompanied by lots of diagrams. The most important
thing to remember when trying to make sense of one of these diagrams is that
two circles in a spherical circle plane that intersect in two points always intersect
transversally at these points – we mentioned this fact before.
Type 1
Two points p and q on the sphere
C + The circles that separate p and q.
C − The circles that miss both p and q and do not separate the two points.
To understand where C + comes from, consider Figure 3. It shows what is happening in the classical model. We start with the two points p and q on the sphere
and their connecting line. Clearly, all planes that intersect the interval pq inside
the sphere give rise to circles on the sphere that separate p and q, and every circle
like this arises from one of these planes. Similarly, all planes that intersect the
interval pq outside the sphere, and intersect the sphere itself nontrivially, give rise
to circles on the sphere that do not separate p and q.
455
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
q
k
p
Figure 3. Two points on the sphere
Proof that we are dealing with a separating set.
Connecting circle: Here we start with three distinct points r, s, and t on the sphere
and we are interested in determining which of the sets C ∪ , C + or C − contains
the connecting circle c of the three points. If the three points r, s, and t are
contained in a circle in C ∪ , then this circle is the connecting circle we are after.
Now assume that r, s and t are not all contained in a circle in C ∪ . In particular,
this implies that all three points are different from p and q. Consider the two
circles (in C ∪ ) determined by r, s, and p, and r, s, and q. If these two circles
coincide, that is, if r, s, p and q are concircular1 , then there are essentially two
different ways in which the four points can be distributed along their connecting
circle; see the first two diagrams in Figure 4. In the first diagram it is clear that
any circle through r and s other than the distinguished one will be contained in
C + (remember that two circles that intersect in two points intersect transversally
in both points). In the second diagram it is clear that any circle through r and s
other than the distinguished one will be contained in C − .
s
p
p
1
r
s
q
p
r
q
r
2
2
1
s
q
Figure 4. The different relative positions of the points p, q, r and s
If the two circles determined by r, s, and p, and r, s, and q are distinct, then we are
in the situation depicted in the last diagram in Figure 4. The two distinguished
1
At this point we could argue that if r, s, p and q are concircular, then r, t, p and q are not,
and we therefore do not have to worry about the case of four concircular points. However, later
on in the proof, when it comes to checking that axiom S1 is satisfied, we will again stumble
across this case. There we won’t have the option of sidestepping this case. Therefore it makes
sense to already deal with it at this point.
456
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
circles dissect the sphere into regions 1 and 2 as indicated. Both regions consist of
two connected components. The point t is contained either in region 1 or region
2. If it is contained in region 1, then, apart from r and s, the connecting circle
we are after is also contained in region 1. This implies that this connecting circle
is contained in C − . Similarly, if t is contained in region 2, we conclude that the
connecting circle we are after is contained in C + .
Touching circle: Here we start with a circle c, a point r on c and a point s
different from r and we are interested in the circle through s that touches c in r.
If either one of the points r or s coincides with p or q, then the touching circle is
contained in C ∪ . Assume that this is not the case. Then we again consider the
three possible configurations of the connecting circles of r, s and p and r, s and q
shown in Figure 4. There are a number of different possible scenarios. If the circle
c touches one of these two distinguished circles, then this distinguished circle is
the touching circle that we are looking for. Otherwise, the circle c intersects the
distinguished circle(s) transversally at r. If this is the case in the first diagram or
the second diagram in Figure 4, then the touching circle is clearly contained in
C + or C − , respectively. If we are dealing with a situation that corresponds to the
last diagram, then, close to r, the circle c is either contained in region 1 or it is
contained in region 2. If the region under discussion is region 1, then except for
the points r and s, the touching circle will be completely contained in region 1,
and hence is an element of C − . Similarly, if, close to r, the circle c is contained
in region 2, then the touching circle we are looking for is also contained in this
region, and hence is an element of C + .
Connectivity: The circles through p form a space homeomorphic to a Möbius
strip and the pencil of circles through p and q forms a non-separating topological
circle in this Möbius strip. This means that any two circles through p that miss q
are connected by a path of circles in the Möbius strip that does not contain any
circle in the pencil. Let us call such a path a missing path. Consider two circles
c0 and c1 in C + and choose points r0 , s0 , t0 on c0 and points r1 , s1 on c1 such
that the connecting circle d0 of r0 , s0 , p and the connecting circle d1 of r1 , s1 , p
do not pass through q. Choose a missing path of circles d : [0, 1] → C, with
d(0) = d0 and d(1) = d1 . Choose paths r : [0, 1] → S2 and s : [0, 1] → S2 such
that r(x), s(x) ∈ d(x) and r(x) 6= s(x), for all x ∈ [0, 1]. This means that, as we
vary x continuously from 0 to 1, the connecting circle d(x) of the points r(x), s(x)
and p and the connecting circle of r(x), s(x) and q will always be distinct, and
we can label regions 1 and 2 as in the right diagram in Figure 4 consistent with
the movement of the connecting circles. Now, choose a path t : [0, 1] → S2 with
t(0) = t0 and t(1) ∈ c1 such that t(x) is contained in region 1 for all x ∈ [0, 1].
Then the moving connecting circle of r(x), s(x) and t(x) is a path completely
contained in C + that connects the two circles c0 and c1 that we started with.
This shows that C + is path-connected. The same argument shows that C − is
path-connected.
For the present type of C ∪ it is obvious that any path in C that connects a
circle in C + with a circle in C − has to contain a circle that contains p or q. Hence
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
457
C ∪ separates C + and C − . However, we also need a formal argument that can be
easily adapted to the other types of separating sets that we will be dealing with in
the following. Here is one such argument: Consider a path of circles c : [0, 1] → C
that starts in a circle c(0) in C + and does not contain any circle in C ∪ . We need to
show that c(1) is also contained in C + . Choose paths r, s, t : [0, 1] → S2 such that
the three points r(x), s(x) and t(x) are distinct for all x ∈ [0, 1] and contained in
the circle c(x). Let us also make sure that the connecting circle of r(0), s(0) and p
does not contain q. Let cp (x) and cq (x) be the connecting circles of r(x), s(x) and
p and r(x), s(x) and q, respectively. Since cp (0) does not contain q, we are in the
situation described by the third diagram in Figure 4, and since c(0) is in C + , the
point t(0) will be contained in region 2. Let us follow our path of circles. If cp (x)
never contains q, then, since our path misses C ∪ , t(x) will always remain in region
2 and, therefore, the whole path is contained in C + . Assume that x1 is the first
time that cp (x) contains q, then, at this point we are necessarily in the situation
depicted in the first diagram in Figure 4, t(x1 ) is off cp (x1 ), and, therefore, c(x1 )
is still in C + (otherwise t(x) would have been squeezed onto the distinguished
circle in the second diagram as the third diagram collapses and c(x1 ) would end
up in C ∪ , which is not possible). If cp (x) returns to not passing through q at a
time x2 , then it necessarily does so as in the third diagram in Figure 4, with t(x2 )
being again contained in region 2, and so on. We conclude that the whole path is
contained in C + . Obviously, this also implies that any path of circles that starts
in C − and misses C ∪ will also be completely contained in C − and that, therefore,
C ∪ separates C + and C − .
Type 2
One point p on the sphere and one inner involution q
C + All circles k such that p and q(k) are in different components of S2 \ k.
To understand where the definition of C + comes from, consider the left diagram
in Figure 5. It shows what is happening in the classical model. To start with, we
have the point p on the sphere, the point q inside the sphere, and their connecting
line. Now we want to find a characterization in terms of the point p and in terms
of the bundle involution associated with q of all those circles that correspond to
planes that intersect the interval pq inside the sphere. Looking at one such circle
k, we see that its image q(k) under this bundle involution is separated (on the
sphere) from p by k. Note that our definition of C + really makes sense because
Lemma 2.1.3 guarantees that for any inner involution q, any circle k that is not
contained in FIX(q) is disjoint from q(k).
458
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
p
r
q
s
2
1
q(k)
q(s)
p
p
q(r)
q(s)
r
2
k
s
q(r)
1
Figure 5. One point on the sphere and one inner involution
Proof that we are dealing with a separating set.
Connecting circle: If the three points r, s, and t are contained in a circle in C ∪ ,
then this circle is the connecting circle we are after. In particular, this is the case
if any one of these three points coincides with p or if any two of the three points
are interchanged by q (see Lemma 2.1.1).
Assume that none of this is the case. Then r and s are contained in exactly
one of the circles fixed by q (this is property (B) in action). If p is contained
in this fixed circle, then the positions of q(r) and q(s) relative to r and s alone
determine whether a circle through r and s other than the distinguished one is
contained in C + or C − . For example, in the case shown in the middle diagram
of Figure 4 any circle through r and s different from the distinguished one clearly
separates its image (through q(r) and q(s)) from p and is therefore contained in
C + . (Note that, by Lemma 2.1.4, the points r and q(r) are separated by s and
q(s) on the distinguished circle and that, by Lemma 2.1.3, any circle through r
and s different from the distinguished one is disjoint from its image under q.)
Now assume that p is not contained in the fixed circle through r and s. Then
the right diagram in Figure 5 shows the circle through p, r, and s, the distinguished
fixed circle through r and s, and the points r, s, q(r), and q(s). This diagram
looks very similar to the third diagram in Figure 4 that we considered for our first
separating set. As in that figure, we label regions 1 and 2. Furthermore, since
q(c) passes through q(r) and q(s), it is easy to verify that t being contained in
region 1 or 2 implies that the connecting circle c of the three points r, s, and t is
contained in C − or C + , respectively.
Touching circle: If either one of the two points r or s coincides with p, then the
touching circle is contained in C ∪ . Assume the two points are interchanged by q.
Then, since by Lemma 2.1.1 every circle through r and s (=q(r)) is contained in
FIX(q), the touching circle will also be contained in C ∪ .
Assume that none of this is the case. Consider the connecting circles of r, s
and p and the unique element of FIX(q) through r and s. If both circles coincide,
that is, if p is contained in the fix-circle, consider again the middle diagram of
Figure 5. If the circle c touches the distinguished circles at r, then the touching
circle we are looking for is the distinguished circle and is therefore contained in C ∪ .
Otherwise, the positions of p, r, s, q(r) and q(s) on the distinguished circle alone
determine whether another circle through r and s is contained in C + or C − .
459
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
If the connecting circle of r, s and p and the unique element of FIX(q) through
r and s do not coincide, then the right diagram of Figure 5 is what we have to
consider again and we can finish our argument as for our type 1 separating set.
Connectivity: Using Lemma 2.5.1, we can adapt the respective arguments for type
1 separating sets to show that both C + and C − are path-connected.
Type 3
Two inner involutions p and q
C + All circles k such that p(k) and q(k) are in different components of S2 \ k.
As in the case of the previous two separating sets, the classical model motivates
the definition of C + ; see the left diagram in Figure 6.
p(k)
p'
p
k
q'
q
1
p(s)
p(r)
r
q(s)
p(s)
r
2
s
q(r)
q(k)
2
p(r)
2
s
q(r)
q(s)
1
Figure 6. Two inner involutions
Sketch of a proof that we are dealing with a separating set.
Connecting circle: If the three points r, s, and t are contained in a circle in C ∪ ,
then this circle is the connecting circle we are after. In particular, this is the case
if any two of these three points are interchanged by p or q. Assume that none of
this is the case. Then r and s are contained in exactly one of the circles fixed by
p and in exactly one of the circles fixed by q. If these two circles coincide, then
we can argue as in the case of our type 2 separating sets that the positions of r,
s, p(r), p(s), q(r), and q(s) on the distinguished circle alone determine whether
another circle through r and s is contained in C + or C − . For example, if these
points are situated as in the middle diagram of Figure 6, then the connecting
circle c necessarily separates its images under p and q and is therefore contained
in C + .
Similarly, if the two fixed circles are distinct, we can argue as in the corresponding case of our type 2 separating set, using the right diagram of Figure 6.
Touching circle & Connectivity: Again it is a straightforward exercise to adapt
the arguments that we used in the case of our type 2 separating set to deal with
this situation.
460
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
Expressing type 3 in terms of type 2
Use the left diagram in Figure 6 to convince yourself that in the classical setting
the line through the two inner points p and q intersects the sphere in two points
p0 and q 0 . In terms of the bundle involutions associated with p and q these two
points on the sphere form the only pair of points that are interchanged by both
involutions. Also, it is easy to see that
C + (p, q) = C + (p0 , q) ∩ C + (q 0 , p)
and that
C − (p, q) = C − (p0 , q) ∪ C − (q 0 , p).
Furthermore, we always have
C ∪ (p, q) = C \ (C + (p, q) ∪ C − (p, q)).
This means that in the classical setting the separating set C ∪ (p, q) of type 3
and its associated sets C + (p, q) and C − (p, q) can be expressed in terms of two
separating sets of type 2 and their associated sets. This relationship generalizes
to our general setting in which both p and q are inner involutions. Note that
Lemma 2.6.1 guarantees the existence of a unique pair of points on the sphere
that are interchanged by both p and q. To figure out which of these points should
be called p0 and which q 0 , consider a circle k in C + . Then, as in the left diagram
of Figure 6,
(O) p0 is the point that is separated from k by p(k) and q 0 is the point that is
separated from k by q(k).
Proposition 3.1. (type 3 via type 2) Let p and q be distinct inner involutions
of a spherical circle plane, and let p0 , q 0 be the unique pair of points on the sphere
interchanged by both involutions and labeled as specified under (O), above. Then
C + (p, q) = C + (p0 , q) ∩ C + (q 0 , p) and C − (p, q) = C − (p0 , q) ∪ C − (q 0 , p).
Proof. We first show that every circle in C + (p, q) separates p0 and q 0 . Consider
a circle that contains p0 . Then the images of this circle under p and q intersect
in q 0 . We conclude that no circle in C + (p, q) contains p0 or q 0 . Let us consider a
circle c in the spherical circle plane that does not contain p0 or q 0 and also does
not separate the two points. Then there is a circle d through p0 and q 0 that does
not intersect c. Since d is fixed by both p and q, both p(c) and q(c) are contained
in the connected component of S2 \ d that c is not contained in. Hence c does not
separate p(c) and q(c) and is therefore not contained in C + (p, q). We conclude
that every circle in C + (p, q) separates p0 and q 0 .
We now need to show that the labeling of p0 and q 0 as specified under (O)
above is independent of the choice of the circle k in C + (p, q). Let l be a second
circle in C + (p, q). Since C + (p, q) is path-connected, there is a path of circles in
C + (p, q) connecting k and l. As we move along this path, the relative positions
of the two points p0 , q 0 , of the moving circle, and its two images cannot change.
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
461
Hence the labeling of p0 and q 0 is independent of the circle k. Furthermore, if k is
any circle in C + (p, q), then p(k) separates p0 from k and q(k) separates q 0 from k
and on the sphere the two points and three curves are therefore separated out in
the following order: p0 , p(k), k, q(k), q 0 .
Now it is clear from the definitions of the C + sets for type 2 and type 3
separating sets that C + (p, q) = C + (p0 , q) ∩ C + (q 0 , p).
Also, if k is a circle in FIX(p)\FIX(q) = FIX(p)\C ∩ (p, q), then there is a path
that ends in k but is otherwise completely contained in C + (p, q). By continuity, it
then follows easily that q(k) is disjoint from k (touching is impossible because this
would give a fixed point of q), and that the two points p0 , q 0 and the two curves
are separated out on the sphere in the following order: p0 , k = p(k), q(k), q 0 .
To finish the proof we need to show that C − (p, q) = C − (p0 , q) ∪ C − (q 0 , p). Let
us first spell out characterizations for the C − sets associated with type 2 and type
3 separating sets. First, C − (p, q) is the set of all circles k such that both p(k) and
q(k) are contained in one connected component of S2 \ k, and C − (p0 , q) is the set
of all circles such that p0 and q(k) are contained in the same connected component
of S2 \ k.
Since C ∪ (p0 , q) contains C ∩ (p, q), a circle k in C − (p0 , q) is not contained in
∩
C (p, q). If k was contained in FIX(p), then p0 , q(k), k would be disjoint and
separated out on the sphere in this order. Since this order does not mesh in
with that derived above for a circle in FIX(p) \ C ∩ (p, q), we conclude that k is
not contained in C ∪ (p, q). Also, since the order p0 , p(k), k, q(k), q 0 is definitely
not present for this circle it is not contained in C + (p, q). Hence we can be sure
that it is contained in C − (p, q). Thus C − (p0 , q) ∪ C − (q 0 , p) ⊂ C − (p, q). Now let
c be a circle in C − (p, q). If c contains q 0 , then q(c) contains p0 and therefore p0
and q(c) are in the same connected component of S2 \ c. Hence, c is contained
in C − (p0 , q). We conclude that if c contains either p0 or q 0 , then it is contained
in C − (p0 , q) ∪ C − (q 0 , p). Assume that c contains neither p0 nor q 0 . Since c is
contained in C − (p, q), both q(c) and p(c) are contained in the same connected
component of S2 \ c. This component also contains either p0 or q 0 . If it contains
p0 , then it is contained in C − (p0 , q), otherwise in C − (q 0 , p). We conclude that c is
contained in C − (p0 , q) ∪ C − (q 0 , p). Hence C − (p, q) ⊂ C − (p0 , q) ∪ C − (q 0 , p).
462
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
Type 4b
One point p on the sphere and one outer involution q, with p ∈
/ fix(q)
Let H + (q) be the connected component of S2 \ fix(q) that contains p. The set C +
generalizes the set of circles in the classical model corresponding to the planes
that intersect the open interval pq in Figure 7. Hence, C + consists of all circles
not in C ∪ belonging to one of the following categories:
• The circles k completely contained in H + (q) such that the connected component of S2 \ k completely contained in H + (q) does not contain the point
p; see the left diagram in Figure 7.
• All circles k intersecting fix(q) in two points such that restricted to H + (q),
q(k) and p are separated by k; see the right diagram in Figure 7.
q(k)
k
k
fix(q)
p
q
H+(q)
p
q
Figure 7. One outer involution q and one point on the sphere p not fixed by the
involution
Proof that we are dealing with a separating set.
Connecting circle: If the three points r, s, and t are contained in a circle in C ∪ ,
then this circle is the connecting circle we are looking for. In particular, this is the
case if any one of these three points coincides with p or if any two are interchanged
by q. Also, if all three points are fixed by q, then their connecting circle is fix(q),
which is contained in C − .
Assume that none of this is the case and that, w.l.o.g., r is not fixed by q.
Then r and s are contained in exactly one circle through p and in exactly one
circle fixed by q. If these two circles coincide, we have to consider a number of
different cases. Since r is not fixed by q, the two points r and q(r) are contained in
different connected components of S2 \fix(q). Figure 8 shows the different possible
positions of the points r, q(r) and p with respect to the distinguished fixed circle
through r and s and the circle fix(q). The gray squares indicated the essentially
different possible positions of the point s. This means that the every one of the
three diagrams corresponds to as many cases as there are gray squares.
463
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
q(r)
r
Figure 8.
coincide
r
p
q(r)
fix(q)
p
The fixed circle through r and s and the circle through r, s and p
For two of the possible positions of s the two rows of diagrams in Figure 9 illustrate
the essentially different ways in which a circle through r and s and its image under
q (drawn thick) can be situated with respect to fix(q) and the distinguished circle.
In the second row we skipped the limiting cases in between the first in second
diagram in which the thick circle through r and s touches fix(q). Similarly, we
skipped the limiting case between the second and third diagram.
q(r)
q(r)
q(r)
s=q(s)
s=q(s)
p
s=q(s)
p
r
r
r
q(r)
q(r)
q(r)
q(s)
q(s)
p
q(s)
s
s
r
p
r
p
s
r
p
Figure 9. Possible locations of the circle through r and s and its image under
q (both drawn thick) for two different positions of r, s, q(r) and q(s) on the
distinguished circle
In all diagrams it is easy to see that any circle through r and s different from
the distinguished circle is contained in C − . This means that in the two cases
under consideration we can decide by just looking at the positions of r, s, q(r),
and q(s) on the distinguished circle alone whether another circle through r and s
is contained in C + or C − . This turns out to be true for all cases corresponding to
464
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
Figure 8. To check this is a (tedious) routine exercise. As usual, it turns out that
all cases that need to be considered are exactly those that pop up in the classical
model and there are no surprises.
Now assume that the circle through r, s and p and the fixed circle through r
and s are different. Again, corresponding to the different relative positions of
the distinguished circles, fix(q), p, r, s, q(r) and q(s) there are lots of different
cases that need to be considered. As above this is a routine exercise and there
are no surprises. We just give the details for one of the possible cases; see the left
diagram in Figure 10.
p
p
1
r
2
s
1
q(r)
q(s)
2
p
r
s
q(r)
q(s)
r
q(r)
s
q(s)
Figure 10.
If t is contained in region 1 or region 2, then no matter whether the connecting
circle intersects fix(q) or not, it will be in contained in C + or C − , respectively. For
example, the middle diagram of Figure 10 shows what things will look like if the
connecting circle is contained in region 2 and does not intersect fix(q). Similarly,
the right diagram shows what happens if t is contained in region 2 and intersects
fix(q) in two points.
Touching and Connectivity are again just variations of what we did for the first
type of separating set that we considered.
Type 5a
One inner involution p and one outer involution q, with fix(q) ∈
FIX(p)
Let H + (q) be any one of the two connected components of S2 \fix(q). C + consists
of all circles k
• that are contained in H + (q), but are not equal to fix(q); or
• that intersect fix(q) in two points and, when restricted to H + (q), separate
q(k) from p(k); see Figure 11 for different instances of k in the classical
setting.
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
H+(q)
465
q'
q(k)
p
p'
k
k
p
fix(q)
q
q
p(k)
Figure 11. One inner involution p and one outer involution q such that the circle
that is pointwise fixed by q is also fixed by p
The proofs that this and the remaining two types of separating sets are really
separating sets are fairly straightforward variations (involving lots of cases) of the
proofs so far and will be omitted here.
Also, in some sense these new separating sets are not really that new because
it turns out (again lots of cases) that they can be reduced, just like the type 3
separating sets (two inner involutions), to type 2 and type 4b separating sets. For
example, for the type of separating set at hand, use the left diagram in Figure 11
to convince yourself that in the classical setting the line through the inner and
outer points p and q intersects the sphere in two points p0 ∈ H + (q) and q 0 ∈
/ H + (q).
In terms of the bundle involutions associated with p and q these two points form
the only pair of points that are interchanged by both involutions. Also, it is easy
to see that
C + (p, q) = C − (q 0 , q) ∩ C − (q 0 , p)
and that
C − (p, q) = C − (p0 , p) ∩ C − (p0 , q).
Furthermore, we always have
C ∪ (p, q) = C \ (C + (p, q) ∪ C − (p, q)).
This relationship generalizes to our general setting. Note that Lemma 2.6.1 guarantees the existence of a unique pair of points on the sphere that are interchanged
by both p and q.
Type 5b
One inner involution p and one outer involution q, fix(q) ∈
/ FIX(p)
It is very complicated to describe C ∪ from scratch. So, we again restrict ourselves
to skipping straight to the end of our arguments and describing this set in terms of
the unique pair of points p0 , q 0 on the sphere that are interchanged by p and q. It
turns out that just like in the classical model only one of the connected components
of S2 \ fix(q) contains some circles in FIX(p). Let us call this component H + (q).
466
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
If p0 , q 0 is the unique pair of points on the sphere that are interchanged by both
p and q, label with p0 the one of the two points that is contained in H + (q) and
q 0 the one that is not; see the left diagram in Figure 12 for the classical setup
that motivates this definition. Now, it is again possible to prove that C + (p, q) =
C − (q 0 , q) ∩ C − (q 0 , p) and C − (p, q) = C − (p0 , p) ∩ C − (p0 , q).
q
q'
q'
p
fix(q)
p'
p'
p
q
H+(q)
Figure 12. One inner involution and one outer involution in general position on
the left, and, on the right, two outer involutions whose pointwise fixed circles are
disjoint
Type 6a
Two outer involutions p and q, with fix(p) ∩ fix(q) = ∅
We again skip straight to the end of our arguments and describe the set C + in
terms of the unique pair of points p0 , q 0 on the sphere that are interchanged by p
and q. To label p0 and q 0 , choose a circle k in between fix(p) and fix(q). Then
p0 is the point that is separated from k by p(k) and q 0 is the point separated
from k by q(k); see the right diagram in Figure 12 for the classical setup that
motivates this definition. Now, C + (p, q) = C − (p0 , q) ∪ C − (q 0 , p) and C − (p, q) =
C + (p0 , q) ∩ C + (q 0 , p).
What about cases 4a, 6b, 6c, 6d, ...?
Why not try and also turn diagrams 4a, 6b, 6c, and 6d in Figure 1 into new
separating sets? It should be possible to come up with further new separating
sets that generalize these cases. However, apart from the messy splitting up into
numerous subcases that outer involutions entail, there are added complications
in all these extra cases. For example, diagrams 6b, 6c and 6d correspond to two
outer involutions whose pointwise fixed circles intersect, and in these cases we do
not know much about the structure of the corresponding set C ∩ to define possible
separating sets in the way we have done so far. It may be possible to overcome
these problems by adding extra assumptions that guarantee that we stay close to
the classical setup, but we have decided not to pursue this any further.
B. Polster, G. F. Steinke: Virtual Points and Separating Sets . . .
467
In this context it is also important to point out again that we are interested in
separating sets mainly because a separating set contained in two spherical circle
planes allows us to combine these two circle planes into a new spherical circle
plane. Sadly, unlike for most of the other types of geometries on s urfaces, we
have not been able to produce an example of any of the new types of separating
sets that is simultaneously contained in two different spherical circle planes. The
closest we got to this goal are the examples of pairs of ovoidal spherical circle
planes constructed in the previous section that share an inner or outer involution.
Also, it is worth pointing out that the derived incidence structure at a point p of
an ovoidal spherical circle plane (basically the set C(p)) is isomorphic to part of
the Euclidean plane. In particular, the derived incidence structure of a ovoidal
flat Möbius plane is isomorphic to the Euclidean plane itself. This means that,
using a suitable homeomorphism of the sphere to itself, we can always arrange
isomorphic copies of any two ovoidal flat Möbius planes such that for one of the
points of the sphere the sets C(p) of both planes coincide. So, the closest we have
been able to come to our goal of finding two spherical circle planes sharing one of
our new separating sets is finding two spherical circle planes that share “half” of
such a separating set.
References
[1] Polster, B.: Separating sets in interpolation and geometry. Aequationes Math.
56 (1998), 201–215.
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0921.41003
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[2] Polster, B.; Steinke, G. F.: Cut and paste in 2-dimensional projective planes
and circle planes. Can. Math. Bull. 38 (1995), 469–480.
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0844.51005
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[3] Polster, B.; Steinke, G. F.: The inner and outer space of 2-dimensional Laguerre planes. J. Aust. Math. Soc., Ser. A 62 (1997), 104–127.
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[4] Polster, B.; Steinke, G. F.: Geometries on Surfaces. Encyclopaedia of Mathematics and its Applications 84, Cambridge University Press, 2001.
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[5] Salzmann, H.; Betten, D.; Grundhöfer, T.; Hähl, H.; Löwen, R.; Stroppel, M.:
Compact projective planes. De Gruyter Expositions in Mathematics 21, de
Gruyter, Berlin 1996.
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[6] Stroppel, M.: A note on Hilbert and Beltrami systems. Result. Math. 24
(1993), 342–347.
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Received August 13, 2006